Dynamical characterization of central sets along filter

TL;DR

This paper extends the concept of central sets in semigroup theory by characterizing -central sets dynamically, using the framework of topological dynamics and filters within the Stone-ech compactification.

Contribution

It introduces a dynamic characterization of -central sets generated by filters in the Stone-ech compactification, generalizing previous algebraic characterizations.

Findings

-central sets are characterized dynamically.

The paper links filter-generated subsemigroups to minimal idempotents.

Provides a new perspective on central sets via topological dynamics.

Abstract

Using the notions of Topological dynamics, H. Furstenberg defined central sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman characterized central sets in terms of algebra of the Stone-\v{C}ech Compactification of discrete semigroup. They found that central sets are the members of the minimal idempotents of $\beta S$, the Stone-\v{C}ech Compactification of a semigroup $\left(S,\cdot\right)$. We know that any closed subsemigroup of $\beta S$ is generated by a filter. We call a set $A$ to be a $\mathcal{F}$-central set if it is a member of a minimal idempotent of a closed subsemigroup of $\beta S$, generated by the filter $\mathcal{F}$. In this article we will characterize the $\mathcal{F}$-central sets dynamically.